Documentation / Guide méthodologique :

The concept of polynomial chaos development relies on the homogeneous chaos theory introduced by Wiener in 1938 [Wiener38] and further developed by Cameron and Martin in 1947 [Cameron47]. Using polynomial chaos (later referred to as PC ) in numerical simulation has been brought back to the light by Ghanem and Spanos in 1991 [Ghanem91]. The basic idea is that any square-integrable function can be written as where are the PC coefficients, is the orthogonal polynomial-basis. The index over which the sum is done, , corresponds to a multi-index whose dimension is equal to the dimension of vector (i.e. ) and whose L1 norm () is the degree of the resulting polynomial. Originally written to deal with normal law, for which the orthogonal basis is Hermite polynomials, this decomposition is now generalised to few other distributions, using other polynomial orthogonal basis (the list of those available in Uranie is shown in Table IV.1).

This decomposition can be helpful in many ways: it can first be used as a surrogate model but it gives also access, through the value of its coefficient, to the sensitivity index (this will be first introduced in Section IV.3.1.2 and further developed in Chapter V).

We'll discuss here a simple example of polynomial chaos development and its implication. In the case where a system is depending on two random variables, and that follow respectively an uniform and normal distribution, giving rise to a single output . Following the remark about square-integrable functions, both inputs can be decomposed on a specific orthogonal polynomial-basis, such as , and , where and are the PC coefficients that respectively multiply the Legendre () and Hermite () polynomials, for the uniform and normal law and where is the multi-index (here of dimension 1) over which the sum is done. These basis are said to be orthogonal because for any degrees and , taking the Legendre case as an example, one can write , for .

It is now possible to write the output, , as a function of these polynomials. For the -Th simulation,

Equation IV.4. Chaos polynomial function

where is the multi-index of dimension 2 () over which the sum is performed. The polynomials are built by tensor products of the inputs basis following the previously defined degree. In the specific case of the simple example discussed here, this leads to a decomposition of the output that can be written as

Equation IV.5.  polynomial chaos decomposition

From this development, it becomes clear that a threshold must be chosen on the order of the polynomials used, as the number of coefficient is growing quickly, following this rule , where is the cut-off chosen on the polynomial degree. In this example, if we choose to use , this leads to only 6 coefficients to be measured: . Their estimation is discussed later.

These coefficients are characterising the surrogate model and can be used, when the inputs are independent, to estimate the corresponding Sobol's coefficients (a deeper discussion about these coefficients and their meaning can be found in Section V.1). For the uniform and normal example, the first order coefficients are respectively given by

whereas the total order coefficients are respectively given by

The complete variance of the output, can also be written as

The regression method is simply based on a least-squares approximation: once the design-of-experiments is done, the vector of output is computed with the code. The regression coefficients are estimated considering that every computed output points can be represented following Equation IV.5. By writing the correspondence matrix and the coefficient-vector , this estimation is just a minimisation of , where, once back to our simple example from Section IV.3.1.2 for illustration purpose,

In order to perform this estimation, it is mandatory to have more points in the design-of-experiments than the number of coefficient to be estimated (in principle, following the rule leads to a safe estimation).

The integration method relies on a more "complex" design-of-experiments. It is indeed recommended to have dedicated design-of-experiments, made with a Smolyak-based algorithms (as the ones cited in Figure IV.5). These design-of-experiments are sparse-grids and usually have a smaller number of points than the regularly-tensorised approaches. In this case, the number of samples has not to be specified by the user. Instead, the argument requested describes the level of the design-of-experiments (which is closely intricated, as the higher the level is, the larger the number of samples is). Once this is done, the calculation is performed as a numerical integration by quadrature methods, which requires a large number of computations.
In the case of Smolyak algorithm, this number can be expressed by the number of dimensions and the requested level as which shows an improvement with respect to the regular tensorised formula for quadrature ().